Page 296 - Advanced Linear Algebra
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280 Advanced Linear Algebra
/# ~ c#Á /$ ~ $ for all $ º#»
#
#
and that
º%Á #»
/% ~ % c #
#
º#Á #»
In the present context, we do not dare divide by º#Á #» , since all vectors are
isotropic. So here is the next-best thing.
=
Definition Let be a nonsingular symplectic geometry over . Let - # = be
nonzero and let - . The map #Á ¢ = ¦ = defined by
#Á ²%³ ~ % b º%Á #»#
is called the symplectic transvection determined by and .
#
Note that if ~ , then #Á ~ and if £ , then #Á is the identity precisely
on the subspace span²#³ of codimension . In the case of a reflection, / # is the
identity precisely on span²#³ and
=~ span ²#³ p span ²#³
is the identity precisely on
However, for a symplectic transvection, #Á
span²#³ (for £ ) but span²#³ span²#³ . Here are the basic properties of
symplectic transvections.
be a symplectic transvection on . Then
Theorem 11.17 Let #Á =
) is a symplectic transformation isometry .
)
(
1 #Á
2 ) #Á ~ if and only if ~ .
)
3 If %# , then #Á ²%³ ~% . For £ , %# if and only if #Á ²%³~% .
)
4 #Á ~ #Á b .
#Á
c
5 ) #Á ~ #Ác .
6 For any symplectic transformation ,
)
c ~ #Á
#Á
)
7 For - i ,
#Á ~ #Á
<
=
<
Note that if is a subspace of and if "Á is a symplectic transvection on ,
then, by definition, "< . However, the formula
"Á ²%³ ~ % b º%Á "»"
also defines a symplectic transvection on , where ranges over . Moreover,
=
=
%
for any ' < , we have "Á '~ ' and so "Á is the identity on < .
